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Prediction of Differential Joule-Thomson Inversion Curves for Cryogens Using Equations of State

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Prediction of Differential Joule-Thomson Inversion Curves for Cryogens Using Equations of State PREDICTION OF DIFFERENTIAL JOULE-THOMSON INVERSION CURVES FOR CRYOGENS USING EQUATIONS OF STATE Nupura Ravishankar1, P.M. Ardhapurkar2, M.D. Atrey3 1Sardar Vallabhbhai National Institute of Technology, Surat – 395 007 2S.S.G.M. College of Engineering, Shegaon – 444 203 3Indian Institute of Technology Bombay, Mumbai – 400 076 In many cryogenic applications, Joule–Thomson (J–T) effect is used to produce low temperatures. The isenthalpic expansion of gas results in lowering of temperature only if the initial temperature of the working fluid is below its characteristic temperature, called the inversion temperature. Inversion curves are helpful in studying the inversion temperatures. The prediction of inversion curves solely depends on the equation of state (EOS) for the working fluid. In the present work, various EOS are explored in order to predict the Joule–Thomson differential inversion curve for various commonly used cryogens viz. nitrogen, argon, carbon dioxide, helium, hydrogen and neon. The widely accepted EOS such as Van der Waals, Redlich–Kwong and Peng–Robinson EOS are used for this purpose. Key words: Inversion curve, Joule Thomson Effect, Van der Waal, Redlich Kwong, Peng Robinson INTRODUCTION Joule–Thomson coefficient, µJT and is given as, The Joule-Thomson (J-T) effect has been widely investigated because of its 흏푻 흁푱푻 = ( ) (1) importance for theoretical and practical 흏푷 푯 purposes. It has many cryogenic applications and is widely used in gas The locus of points for which μJT =0 is liquefaction. The effect describes called the differential inversion curve. The the temperature change of a gas or liquid inversion curve divides the p-T plane into when it is forced through a valve or porous two zones as shown in Figure 1. The J–T plug under adiabatic conditions. coefficient is positive inside the inversion The rate of change of temperature with curve, while it is negative outside the curve. respect to pressure in a J-T process (that is, at constant enthalpy) is called the differential Avraham [7], developed a new procedure to investigate the capability of an EOS to predict an experimental inversion curve in a mathematical way. There are many EOS available in the literature which are claimed to be accurate for specific fluids and for a range of operating conditions only. Therefore, selection of suitable EOS to predict accurate inversion curve for the given cryogen is an important step in the simulation or design Figure 1: J-T inversion curve dividing the p-T plane into studies of the cryogenic equipment involving two zones. J–T expansion. The prediction of inversion curves The present work focuses on the depends solely on the equation of state applicability of various EOS for predicting (EOS) that is used for a given working fluid, the J-T differential inversion curve for which operates in a particular p-T region. various commonly used cryogens, Nitrogen This prediction has been studied by various (N2), viz. Argon (Ar) , Carbon dioxide (CO2), researchers [1-2]. Miller [3] suggested that it Helium (He), Neon (Ne) and Hydrogen (H2). would be desirable for constructors of new The paper presents the calculations of EOS to include a comparison of the differential inversion curves and their predicted inversion curve with an comparison with the experimental data experimental one. Prior to 1970, J-T available in the literature. inversion curves were plotted only for a few EOS namely, Van der Waals, Dieterici, THEORY AND ANALYSIS Lennard-Jones-Devonshire, and DeBoer- Michels EOS [4]. Miller calculated inversion The differential inversion curve is defined as curves for the Redlich-Kwong (RK) and the locus of all points where the Joule- Martin EOS, while Juris and Wenzel Thomson coefficient μJT vanishes. It has a calculated inversion curves for the virial, maximum inversion temperature, i.e., the Berthelot, Beattie-Bridgeman, Benedict- maximum temperature when the pressure is Webb-Rubin, Redlich-Kwong, and Martin- zero. For cooling, it is necessary for the gas Hou EOS. to be below this temperature and inside the Dilay and Heldermann [5] used four inversion curve. recently published EOS to calculate the J-T In the present work, three equations of inversion curves; namely, Soave Redlich state, i.e, Van der Waals, RK, PR have been Kwong, Peng Robinson (PR), Perturbed analysed to understand their applicability to hard chain and Lee Kesler EOS. They predict the inversion curves. Eqn. (2) gives concluded that none of the EOS were able the condition for differential inversion curve. to predict the entire inversion curve 휕푉 푇 ( ) − 푉 accurately; however the Lee Kesler EOS 휕푇 휕푇 푃 휇퐽푇 = ( ) = = 0 (2) gave the best overall prediction. The 휕푃 퐻 퐶푝 obtained inversion curves were then compared with the inversion curve developed by Gunn Chueh, and Prausnitz 휕푉 Thus, 푇 ( ) − 푉 = 0 (3) [2] from the data available for simple fluids. 휕푇 푃 Maytal and Shavit [6] developed the integral Which can also be written as, inversion curve concept, involving the locus of all points with a vanishing integral Joule Thomson effect, ∆Th and isothermal 휕푃 휕푃 휇퐽푇 = 푇 ( ) + 푉 ( ) = 0 (4) enthalpy change, ΔHT. Wisniack and 휕푇 푉 휕푉 푇 The condition so derived in Eqn. (4) is In the reduced form it is: applied to the above mentioned EOS based on which the inversion curves are developed 푇푟 Ω푎 for various gases. The following section 푃푟 = − 2 (9) Ω푏(푥−1) Ω √푇푟푥(푥+1) gives these details. 푏 Van der Waals equation of state Where x is the reduced volume and Ωa The Van der Waals EOS is expressed and Ωb are constants with values as given, in Eqn. (5): Ωa =0.4275 푎 (푃 + ) (푉 − 푏) = 푅푇 (5) 푉2 푚 푚 Ωb =0.08664 The Van der Waals constant ‘a’ is a By applying the equation for differential measure of strength of the Van der Waals condition, Tr can be written as, force between the molecules of the gas, while the Van der Waals constant ‘b’ 3 2 ⁄2 Ω푎(5푥+3)(푥−1) 푇푟 = 2 (10) represents effective volume of the gas Ω푏2푥(푥+1) molecules. The equation can be written in its Peng Robinson equation of state reduced form, as given in Eqn. (6), using The PR EOS is as given in Eqn. (11) pressure, temperature and volume at the critical point. The reduced pressure (Pr), 푅푇 푎훼 푃 = − 2 2 (11) temperature (Tr) and volume (Vr) are given 푉−푏 푉 +2푏푉−푏 as P/Pc, V/Vc and T/Tc respectively. Where parameters a, b and α are given 3 1 8푇푟 by: (푃푟 + 2) (푉푟 − ) = (6) 푉푟 3 3 2 2 0.457235푅 푇푐 By applying the equation for differential 푎 = 푝푐 condition, Pr can be written as, 0.077796푅푇푐 [7] 푏 = 푃푟 = 24√3푇푟 − 12푇푟 − 27 푝푐 Redlich Kwong Equation of State 2 The RK EOS [5] is given as: 훼 = (1 + 푘(1 − √푇푟)) 푅푇 푎 푃 = − (8) Where k is: 푉−푏 √푇푉(푉+푏) 푘 = 0.37464 + 1.54226휔 − 0.26992휔2 Where parameters ‘a’ and ‘b’ have the same physical significance as in Van der Where ω is the acentric factor, which is Waals EOS. However, their values are a conceptual number useful in the different which are given as below: description of matter. It is a measure of the non sphericity of molecules. 5 2 ⁄2 In the present work, ω has been 0.4275푅 푇푐 푎 = assumed to be zero, which leads to the 푃 푐 value of k=0.37464. 0.08664푅푇 푏 = 푐 푃푐 In the reduced form, the PR EOS is: are used and results are compared with the experimental data [5] for the inversion curve. 2 푇푟 Ω푎[1+푘(1−√푇푟)] As is evident from the figure, the 푃푟 = − 2 2 (12) Ω푏(푥−1) Ω푏(푥 +2푥−1) inversion curves appear to be qualitatively Where x is the reduced volume similar to each other. But when studied closely, they give varied data of the Ωa =0.457235 inversion temperatures. The curves are divided into two temperature zones - a high Ωb =0.077796 temperature branch and a low temperature branch. The low temperature branch of When the equation for differential these curves closely match with the condition is applied, Tr can be written as, experimental curve. However, at higher temperatures, the curves deviate 2 2 2 2 2 푇푟[Ω푎(푥 − 1) (푥 + 1)푘 −Ω푏(푥 + 2푥 − 1) ] significantly from the experimental data. It is + seen in Figure 2, that Van der Waals EOS √푇 [−Ω (푥 − 1)2푘(푘 + 1)(3푥2 + 2푥 + 1)] + gives good prediction of the low temperature 푟 푎 branch of the inversion curve below 2 bar, 2Ω 푥(푥 + 1)(푥 − 1)2(1 + 푘)2 = 0 (13) 푎 whereas RK and PR EOS give good prediction of the same upto 8 bar. The peak The above derived expressions in pressure also varies for each EOS, being terms of reduced pressure, temperature and less than the experimental data for Van der volume are solved to obtain various values Waals EOS, and more than that for RK and of Pr for different values of Tr to get various PR EOS. It is 9 bar for Van der Waals, 10.8 inversion curves. These reduced forms of bar for RK, 13 for PR and 11.5 bar for the the equations are solved in MATLAB®. For experimental inversion curve. Thus, the high the RK and PR EOS, for different values of temperature branch and the peak pressure reduced volume x, the corresponding values of the inversion curve prove to be sensitive of reduced temperature T are computed r to the EOS that is used. The RK equation of using equations (10) and (13) respectively.
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